The obvious question coming up is "pay to who ?" there is nobody around to collect money...
One potential answer is pay to the Sheep in the next chapter: in chapter V Alice can buy two eggs or one. She decides to buy one and so she meets later Humpty Dumpty. Was the option of two hinting back to the Tweedle brothers ?
"The first thing in a visit is to say "How d'ye do?" and shake hands!"
The Tweedle brothers weren't too polite too since they welcomed Alice by teasing her. But the fact that one is bad- mannered doesn't give the other to be bad- mannered too, contrariwise it emphasizes it's duty to by polite. Please note too that this "How do you do" part is too late now since it's far after the introduction.
"Tweedledee was trying his best to fold up the umbrella, with himself in it"
If Tweedledee could make it and fold up the umbrella, with himself in it his brother's anger and annoyance would be out, so he's safe. It's the same logic used few lines before when Tweedledum spread the umbrella and they said that under the umbrella it's not raining and outside it may rain, they had no objection.
The fight between the Tweedle brothers
This fight, like all kid's fights is lots of ceremony and very little real battles (thinking it over, it's true for most colonial wars too of the nations). In this case, I find it more like children war since the black cloud that interrupted the fight can be the nanny or any other adult that makes the kids run together united by the "fear" from her- anyone that had children saw such a behavior many times.